Problem 29 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ g(x)=5-5 x $$

Step-by-Step Solution

Answer

Slope: -5, Y-intercept: 5. Line through (0, 5) and (1, 0).

Step 1: Identifying the Function

We start with the given linear function: \( g(x) = 5 - 5x \). This can be rewritten in the slope-intercept form \( y = mx + b \). Compare it with \( g(x) = -5x + 5 \). Here, \( m \), the slope, is \(-5\) and \( b \), the y-intercept, is \(5\).

Step 2: Plotting the Y-Intercept

The y-intercept \( b \) is \(5\). To plot this, we find the point where the graph crosses the y-axis, which is at \((0, 5)\). Place a point on the graph at \((0, 5)\).

Step 3: Using the Slope

The slope \( m \) is \(-5\), which means for each unit increase in \( x \), \( y \) decreases by \(5\) units. From the point \((0, 5)\), move right \(1\) unit along the x-axis, then move \(5\) units down along the y-axis to reach another point \((1, 0)\). Plot this point.

Step 4: Drawing the Line

Draw a straight line through the points \((0, 5)\) and \((1, 0)\). This line represents the graph of the function \( g(x) = 5 - 5x \).

Step 5: Verifying the Graph

Check a few additional points on the line. For instance, at \( x = 2 \), \( g(2) = 5 - 5(2) = -5 \). This gives the point \((2, -5)\), which should lie on the line if plotted.

Key Concepts

Exploring the Concept of Slope

In the realm of linear functions, the slope is a crucial component that describes how steep the line is. It also reflects the direction the line travels across the graph. Mathematically, the slope is represented by the letter \( m \) in the slope-intercept form of the line's equation, \( y = mx + b \).

• A positive slope means the line rises from left to right.
• A negative slope indicates the line falls from left to right.
• A zero slope signifies a horizontal line, while an undefined slope points to a vertical line.

In our example, with the function \( g(x) = 5 - 5x \), we identified the slope as \( -5 \). This tells us that the line will decrease by 5 units vertically for every one unit it moves horizontally to the right. Understanding the slope's effect is vital for predicting and plotting the behavior of linear functions on a graph.

Understanding Y-Intercept

In any linear function, the y-intercept refers to the point where the line crosses the y-axis. It is represented by the letter \( b \) in the equation \( y = mx + b \). Since this crossing occurs at the y-axis, the x-coordinate of this point is always zero.
For our function, \( g(x) = 5 - 5x \), we know the y-intercept is \( 5 \). This manifests on the graph as the coordinate point \((0, 5)\). We identify this point as the starting spot from which the graph begins. The y-intercept is a cornerstone for drawing the line of the function, as it gives us a precise initial point from which to use the slope to find additional points on the graph.

Steps to Graphing Linear Functions

Graphing a linear function requires using the slope and y-intercept to accurately portray the line on a coordinate plane. Here is a step-by-step method to graph a function such as \( g(x) = 5 - 5x \).

• Identify the y-intercept: Start by plotting the y-intercept on the graph. For this example, plot the point \((0, 5)\).
• Apply the slope: From the y-intercept, use the slope \( -5 \). Move horizontally one unit to the right, then vertically down five units to plot another point, such as \((1, 0)\).
• Draw the line: Connect these points with a ruler to form a straight line extending in both directions.
• Verify with additional points: Check the accuracy of your graph by calculating another point, for example, \( x = 2 \) results in \( g(2) = -5 \), giving point \((2, -5)\).

Understanding and applying these steps ensures that the process of graphing linear functions becomes intuitive and manageable. This foundational knowledge empowers students to tackle more complex graphing tasks with confidence.